The spatial curvature of the Universe (hereafter denoted as
its contribution relative to the present-day critical density,
\(\Omega _K\) )
is one of the most fundamental quantities in an isotropic and homogeneous universe in the context of General Relativity [1]}.
The curvature also
has a close connection
to the physics of the early universe.
An inflationary universe scenario predicts that the apparent curvature, inferred from an observable universe, should be close to
a flat geometry (\(\Omega _K\approx 0\) ), even if its exact value is non-zero [2]}, [3]}.
If the universe arose form the decay of a false vacuum via quantum tunneling, as inspired by the landscape of string cosmology vacua,
it leads to an open-geometry universe (\(\Omega _K>0\) )
[4]}, [5]}. Depending on the details of the early universe physics, the curvature can be large enough, such as
\(|\Omega _K|\simeq [10^{-4},10^{-2}]\) , to be measurable
by cosmological observations [6]}, [7]}If the curvature is as small as \(|\Omega _K|\sim 10^{-5}\) ,
we cannot distinguish between the global curvature and the primordial curvature “perturbations”. Hence, the target goal for a hunt of the non-zero curvature is in the range \(|\Omega _K|\gtrsim \mbox{a few}\times 10^{-5}\) ..
Therefore, an observational exploration of the curvature is an important direction to pursue with ongoing and upcoming cosmology
datasets [8]}, [9]}.
